# Polynomial Long Division Calculator

## Perform the long division of polynomials step by step

The calculator will perform the long division of polynomials, with steps shown.

Related calculators: Synthetic Division Calculator, Long Division Calculator

Divide (dividend):

By (divisor):

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### Solution

Your input: find $\frac{5 x^{9} - 4 x^{2} + 2}{5 x + 10}$ using long division.

Write the problem in the special format (missed terms are written with zero coefficients):

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}\phantom{x^{8}}&\phantom{- 2 x^{7}}&\phantom{+4 x^{6}}&\phantom{- 8 x^{5}}&\phantom{+16 x^{4}}&\phantom{- 32 x^{3}}&\phantom{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\5 x+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}\end{array}&\begin{array}{c}\end{array}\end{array}$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $\frac{5 x^{9}}{5 x}=x^{8}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $x^{8}\left(5 x+10\right)=5 x^{9}+10 x^{8}$.

Subtract the dividend from the obtained result: $\left(5 x^{9}- 4 x^{2}+2\right)-\left(5 x^{9}+10 x^{8}\right)=- 10 x^{8}- 4 x^{2}+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}\color{Green}{x^{8}}&\phantom{- 2 x^{7}}&\phantom{+4 x^{6}}&\phantom{- 8 x^{5}}&\phantom{+16 x^{4}}&\phantom{- 32 x^{3}}&\phantom{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}\color{Green}{5 x^{9}}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\frac{\color{Green}{5 x^{9}}}{\color{Magenta}{5 x}}=\color{Green}{x^{8}}\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\end{array}\end{array}$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 10 x^{8}}{5 x}=- 2 x^{7}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $- 2 x^{7}\left(5 x+10\right)=- 10 x^{8}- 20 x^{7}$.

Subtract the remainder from the obtained result: $\left(- 10 x^{8}- 4 x^{2}+2\right)-\left(- 10 x^{8}- 20 x^{7}\right)=20 x^{7}- 4 x^{2}+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&\color{Chocolate}{- 2 x^{7}}&\phantom{+4 x^{6}}&\phantom{- 8 x^{5}}&\phantom{+16 x^{4}}&\phantom{- 32 x^{3}}&\phantom{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&\color{Chocolate}{- 10 x^{8}}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\frac{\color{Chocolate}{- 10 x^{8}}}{\color{Magenta}{5 x}}=\color{Chocolate}{- 2 x^{7}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\end{array}\end{array}$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{20 x^{7}}{5 x}=4 x^{6}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $4 x^{6}\left(5 x+10\right)=20 x^{7}+40 x^{6}$.

Subtract the remainder from the obtained result: $\left(20 x^{7}- 4 x^{2}+2\right)-\left(20 x^{7}+40 x^{6}\right)=- 40 x^{6}- 4 x^{2}+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&- 2 x^{7}&\color{Fuchsia}{+4 x^{6}}&\phantom{- 8 x^{5}}&\phantom{+16 x^{4}}&\phantom{- 32 x^{3}}&\phantom{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&\color{Fuchsia}{20 x^{7}}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}}\\\frac{\color{Fuchsia}{20 x^{7}}}{\color{Magenta}{5 x}}=\color{Fuchsia}{4 x^{6}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\end{array}\end{array}$

Step 4

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 40 x^{6}}{5 x}=- 8 x^{5}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $- 8 x^{5}\left(5 x+10\right)=- 40 x^{6}- 80 x^{5}$.

Subtract the remainder from the obtained result: $\left(- 40 x^{6}- 4 x^{2}+2\right)-\left(- 40 x^{6}- 80 x^{5}\right)=80 x^{5}- 4 x^{2}+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&- 2 x^{7}&+4 x^{6}&\color{Chartreuse}{- 8 x^{5}}&\phantom{+16 x^{4}}&\phantom{- 32 x^{3}}&\phantom{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&\color{Chartreuse}{- 40 x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&-\phantom{- 40 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&- 80 x^{5}\\\hline\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}}\\\frac{\color{Chartreuse}{- 40 x^{6}}}{\color{Magenta}{5 x}}=\color{Chartreuse}{- 8 x^{5}}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Chartreuse}{- 8 x^{5}}\left(\color{Magenta}{5 x}+10\right)=- 40 x^{6}- 80 x^{5}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\end{array}\end{array}$

Step 5

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{80 x^{5}}{5 x}=16 x^{4}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $16 x^{4}\left(5 x+10\right)=80 x^{5}+160 x^{4}$.

Subtract the remainder from the obtained result: $\left(80 x^{5}- 4 x^{2}+2\right)-\left(80 x^{5}+160 x^{4}\right)=- 160 x^{4}- 4 x^{2}+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&\color{OrangeRed}{+16 x^{4}}&\phantom{- 32 x^{3}}&\phantom{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&-\phantom{- 40 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&- 80 x^{5}\\\hline\phantom{\enclose{longdiv}{}}&&&&\color{OrangeRed}{80 x^{5}}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&-\phantom{80 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+160 x^{4}\\\hline\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chartreuse}{- 8 x^{5}}\left(\color{Magenta}{5 x}+10\right)=- 40 x^{6}- 80 x^{5}}\\\frac{\color{OrangeRed}{80 x^{5}}}{\color{Magenta}{5 x}}=\color{OrangeRed}{16 x^{4}}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{OrangeRed}{16 x^{4}}\left(\color{Magenta}{5 x}+10\right)=80 x^{5}+160 x^{4}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\end{array}\end{array}$

Step 6

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 160 x^{4}}{5 x}=- 32 x^{3}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $- 32 x^{3}\left(5 x+10\right)=- 160 x^{4}- 320 x^{3}$.

Subtract the remainder from the obtained result: $\left(- 160 x^{4}- 4 x^{2}+2\right)-\left(- 160 x^{4}- 320 x^{3}\right)=320 x^{3}- 4 x^{2}+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&\color{DarkMagenta}{- 32 x^{3}}&\phantom{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&-\phantom{- 40 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&- 80 x^{5}\\\hline\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&-\phantom{80 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+160 x^{4}\\\hline\phantom{\enclose{longdiv}{}}&&&&&\color{DarkMagenta}{- 160 x^{4}}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&&-\phantom{- 160 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&- 320 x^{3}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&320 x^{3}&- 4 x^{2}&+0 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chartreuse}{- 8 x^{5}}\left(\color{Magenta}{5 x}+10\right)=- 40 x^{6}- 80 x^{5}}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{OrangeRed}{16 x^{4}}\left(\color{Magenta}{5 x}+10\right)=80 x^{5}+160 x^{4}}\\\frac{\color{DarkMagenta}{- 160 x^{4}}}{\color{Magenta}{5 x}}=\color{DarkMagenta}{- 32 x^{3}}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{DarkMagenta}{- 32 x^{3}}\left(\color{Magenta}{5 x}+10\right)=- 160 x^{4}- 320 x^{3}\\\phantom{320 x^{3}- 4 x^{2}+0 x+2}\end{array}\end{array}$

Step 7

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{320 x^{3}}{5 x}=64 x^{2}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $64 x^{2}\left(5 x+10\right)=320 x^{3}+640 x^{2}$.

Subtract the remainder from the obtained result: $\left(320 x^{3}- 4 x^{2}+2\right)-\left(320 x^{3}+640 x^{2}\right)=- 644 x^{2}+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&- 32 x^{3}&\color{Brown}{+64 x^{2}}&\phantom{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&-\phantom{- 40 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&- 80 x^{5}\\\hline\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&-\phantom{80 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+160 x^{4}\\\hline\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&&-\phantom{- 160 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&- 320 x^{3}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&\color{Brown}{320 x^{3}}&- 4 x^{2}&+0 x&+2\\&&&&&&-\phantom{320 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&320 x^{3}&+640 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&- 644 x^{2}&+0 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chartreuse}{- 8 x^{5}}\left(\color{Magenta}{5 x}+10\right)=- 40 x^{6}- 80 x^{5}}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{OrangeRed}{16 x^{4}}\left(\color{Magenta}{5 x}+10\right)=80 x^{5}+160 x^{4}}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{DarkMagenta}{- 32 x^{3}}\left(\color{Magenta}{5 x}+10\right)=- 160 x^{4}- 320 x^{3}}\\\frac{\color{Brown}{320 x^{3}}}{\color{Magenta}{5 x}}=\color{Brown}{64 x^{2}}\\\phantom{320 x^{3}- 4 x^{2}+0 x+2}\\\color{Brown}{64 x^{2}}\left(\color{Magenta}{5 x}+10\right)=320 x^{3}+640 x^{2}\\\phantom{- 644 x^{2}+0 x+2}\end{array}\end{array}$

Step 8

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{- 644 x^{2}}{5 x}=- \frac{644 x}{5}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $- \frac{644 x}{5}\left(5 x+10\right)=- 644 x^{2}- 1288 x$.

Subtract the remainder from the obtained result: $\left(- 644 x^{2}+2\right)-\left(- 644 x^{2}- 1288 x\right)=1288 x+2$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&- 32 x^{3}&+64 x^{2}&\color{DarkCyan}{- \frac{644 x}{5}}&\phantom{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&-\phantom{- 40 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&- 80 x^{5}\\\hline\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&-\phantom{80 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+160 x^{4}\\\hline\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&&-\phantom{- 160 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&- 320 x^{3}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&320 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&&&-\phantom{320 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&320 x^{3}&+640 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&\color{DarkCyan}{- 644 x^{2}}&+0 x&+2\\&&&&&&&-\phantom{- 644 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&&- 644 x^{2}&- 1288 x\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&&1288 x&+2\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chartreuse}{- 8 x^{5}}\left(\color{Magenta}{5 x}+10\right)=- 40 x^{6}- 80 x^{5}}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{OrangeRed}{16 x^{4}}\left(\color{Magenta}{5 x}+10\right)=80 x^{5}+160 x^{4}}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{DarkMagenta}{- 32 x^{3}}\left(\color{Magenta}{5 x}+10\right)=- 160 x^{4}- 320 x^{3}}\\\phantom{320 x^{3}- 4 x^{2}+0 x+2}\\\phantom{320 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Brown}{64 x^{2}}\left(\color{Magenta}{5 x}+10\right)=320 x^{3}+640 x^{2}}\\\frac{\color{DarkCyan}{- 644 x^{2}}}{\color{Magenta}{5 x}}=\color{DarkCyan}{- \frac{644 x}{5}}\\\phantom{- 644 x^{2}+0 x+2}\\\color{DarkCyan}{- \frac{644 x}{5}}\left(\color{Magenta}{5 x}+10\right)=- 644 x^{2}- 1288 x\\\phantom{1288 x+2}\end{array}\end{array}$

Step 9

Divide the leading term of the obtained remainder by the leading term of the divisor: $\frac{1288 x}{5 x}=\frac{1288}{5}$.

Write down the calculated result in the upper part of the table.

Multiply it by the divisor: $\frac{1288}{5}\left(5 x+10\right)=1288 x+2576$.

Subtract the remainder from the obtained result: $\left(1288 x+2\right)-\left(1288 x+2576\right)=-2574$.

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}x^{8}&- 2 x^{7}&+4 x^{6}&- 8 x^{5}&+16 x^{4}&- 32 x^{3}&+64 x^{2}&- \frac{644 x}{5}&\color{BlueViolet}{+\frac{1288}{5}}&\phantom{+2}\end{array}&\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}5 x^{9}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&- 10 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&20 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&-\phantom{- 40 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&- 80 x^{5}\\\hline\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&-\phantom{80 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+160 x^{4}\\\hline\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&&-\phantom{- 160 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&- 320 x^{3}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&320 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&&&-\phantom{320 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&320 x^{3}&+640 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&- 644 x^{2}&+0 x&+2\\&&&&&&&-\phantom{- 644 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&&- 644 x^{2}&- 1288 x\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&&\color{BlueViolet}{1288 x}&+2\\&&&&&&&&-\phantom{1288 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&&&1288 x&+2576\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&&&\color{SaddleBrown}{-2574}\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Chartreuse}{- 8 x^{5}}\left(\color{Magenta}{5 x}+10\right)=- 40 x^{6}- 80 x^{5}}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{OrangeRed}{16 x^{4}}\left(\color{Magenta}{5 x}+10\right)=80 x^{5}+160 x^{4}}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{DarkMagenta}{- 32 x^{3}}\left(\color{Magenta}{5 x}+10\right)=- 160 x^{4}- 320 x^{3}}\\\phantom{320 x^{3}- 4 x^{2}+0 x+2}\\\phantom{320 x^{3}- 4 x^{2}+0 x+2}\\\phantom{\color{Brown}{64 x^{2}}\left(\color{Magenta}{5 x}+10\right)=320 x^{3}+640 x^{2}}\\\phantom{- 644 x^{2}+0 x+2}\\\phantom{- 644 x^{2}+0 x+2}\\\phantom{\color{DarkCyan}{- \frac{644 x}{5}}\left(\color{Magenta}{5 x}+10\right)=- 644 x^{2}- 1288 x}\\\frac{\color{BlueViolet}{1288 x}}{\color{Magenta}{5 x}}=\color{BlueViolet}{\frac{1288}{5}}\\\phantom{1288 x+2}\\\color{BlueViolet}{\frac{1288}{5}}\left(\color{Magenta}{5 x}+10\right)=1288 x+2576\\\phantom{-2574}\end{array}\end{array}$

Since the degree of the remainder is less than the degree of the divisor, then we are done.

The resulting table is shown once more:

$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{5 x}+10}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrrrrrr}\color{Green}{x^{8}}&\color{Chocolate}{- 2 x^{7}}&\color{Fuchsia}{+4 x^{6}}&\color{Chartreuse}{- 8 x^{5}}&\color{OrangeRed}{+16 x^{4}}&\color{DarkMagenta}{- 32 x^{3}}&\color{Brown}{+64 x^{2}}&\color{DarkCyan}{- \frac{644 x}{5}}&\color{BlueViolet}{+\frac{1288}{5}}&\phantom{+2}\end{array}&Hints\\\color{Magenta}{5 x}+10&\phantom{-}\enclose{longdiv}{\begin{array}{cccccccccc}\color{Green}{5 x^{9}}&+0 x^{8}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\end{array}}&\frac{\color{Green}{5 x^{9}}}{\color{Magenta}{5 x}}=\color{Green}{x^{8}}\\\phantom{\color{Magenta}{5 x}+10}&\begin{array}{rrrrrrrrrr}-\phantom{5 x^{9}}&\phantom{+0 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}5 x^{9}&+10 x^{8}\\\hline\phantom{\enclose{longdiv}{}}&\color{Chocolate}{- 10 x^{8}}&+0 x^{7}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&-\phantom{- 10 x^{8}}&\phantom{+0 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&- 10 x^{8}&- 20 x^{7}\\\hline\phantom{\enclose{longdiv}{}}&&\color{Fuchsia}{20 x^{7}}&+0 x^{6}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&-\phantom{20 x^{7}}&\phantom{+0 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&20 x^{7}&+40 x^{6}\\\hline\phantom{\enclose{longdiv}{}}&&&\color{Chartreuse}{- 40 x^{6}}&+0 x^{5}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&-\phantom{- 40 x^{6}}&\phantom{+0 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&- 40 x^{6}&- 80 x^{5}\\\hline\phantom{\enclose{longdiv}{}}&&&&\color{OrangeRed}{80 x^{5}}&+0 x^{4}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&-\phantom{80 x^{5}}&\phantom{+0 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&80 x^{5}&+160 x^{4}\\\hline\phantom{\enclose{longdiv}{}}&&&&&\color{DarkMagenta}{- 160 x^{4}}&+0 x^{3}&- 4 x^{2}&+0 x&+2\\&&&&&-\phantom{- 160 x^{4}}&\phantom{+0 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&- 160 x^{4}&- 320 x^{3}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&\color{Brown}{320 x^{3}}&- 4 x^{2}&+0 x&+2\\&&&&&&-\phantom{320 x^{3}}&\phantom{- 4 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&320 x^{3}&+640 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&\color{DarkCyan}{- 644 x^{2}}&+0 x&+2\\&&&&&&&-\phantom{- 644 x^{2}}&\phantom{+0 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&&- 644 x^{2}&- 1288 x\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&&\color{BlueViolet}{1288 x}&+2\\&&&&&&&&-\phantom{1288 x}&\phantom{+2}\\\phantom{\enclose{longdiv}{}}&&&&&&&&1288 x&+2576\\\hline\phantom{\enclose{longdiv}{}}&&&&&&&&&\color{SaddleBrown}{-2574}\end{array}&\begin{array}{c}\phantom{5 x^{9}+0 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Green}{x^{8}}\left(\color{Magenta}{5 x}+10\right)=5 x^{9}+10 x^{8}\\\frac{\color{Chocolate}{- 10 x^{8}}}{\color{Magenta}{5 x}}=\color{Chocolate}{- 2 x^{7}}\\\phantom{- 10 x^{8}+0 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Chocolate}{- 2 x^{7}}\left(\color{Magenta}{5 x}+10\right)=- 10 x^{8}- 20 x^{7}\\\frac{\color{Fuchsia}{20 x^{7}}}{\color{Magenta}{5 x}}=\color{Fuchsia}{4 x^{6}}\\\phantom{20 x^{7}+0 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Fuchsia}{4 x^{6}}\left(\color{Magenta}{5 x}+10\right)=20 x^{7}+40 x^{6}\\\frac{\color{Chartreuse}{- 40 x^{6}}}{\color{Magenta}{5 x}}=\color{Chartreuse}{- 8 x^{5}}\\\phantom{- 40 x^{6}+0 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{Chartreuse}{- 8 x^{5}}\left(\color{Magenta}{5 x}+10\right)=- 40 x^{6}- 80 x^{5}\\\frac{\color{OrangeRed}{80 x^{5}}}{\color{Magenta}{5 x}}=\color{OrangeRed}{16 x^{4}}\\\phantom{80 x^{5}+0 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{OrangeRed}{16 x^{4}}\left(\color{Magenta}{5 x}+10\right)=80 x^{5}+160 x^{4}\\\frac{\color{DarkMagenta}{- 160 x^{4}}}{\color{Magenta}{5 x}}=\color{DarkMagenta}{- 32 x^{3}}\\\phantom{- 160 x^{4}+0 x^{3}- 4 x^{2}+0 x+2}\\\color{DarkMagenta}{- 32 x^{3}}\left(\color{Magenta}{5 x}+10\right)=- 160 x^{4}- 320 x^{3}\\\frac{\color{Brown}{320 x^{3}}}{\color{Magenta}{5 x}}=\color{Brown}{64 x^{2}}\\\phantom{320 x^{3}- 4 x^{2}+0 x+2}\\\color{Brown}{64 x^{2}}\left(\color{Magenta}{5 x}+10\right)=320 x^{3}+640 x^{2}\\\frac{\color{DarkCyan}{- 644 x^{2}}}{\color{Magenta}{5 x}}=\color{DarkCyan}{- \frac{644 x}{5}}\\\phantom{- 644 x^{2}+0 x+2}\\\color{DarkCyan}{- \frac{644 x}{5}}\left(\color{Magenta}{5 x}+10\right)=- 644 x^{2}- 1288 x\\\frac{\color{BlueViolet}{1288 x}}{\color{Magenta}{5 x}}=\color{BlueViolet}{\frac{1288}{5}}\\\phantom{1288 x+2}\\\color{BlueViolet}{\frac{1288}{5}}\left(\color{Magenta}{5 x}+10\right)=1288 x+2576\\\phantom{-2574}\end{array}\end{array}$

Therefore, $\frac{5 x^{9} - 4 x^{2} + 2}{5 x + 10}=x^{8} - 2 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} - 32 x^{3} + 64 x^{2} - \frac{644 x}{5} + \frac{1288}{5}+\frac{-2574}{5 x + 10}$

Answer: $\frac{5 x^{9} - 4 x^{2} + 2}{5 x + 10}=x^{8} - 2 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} - 32 x^{3} + 64 x^{2} - \frac{644 x}{5} + \frac{1288}{5}+\frac{-2574}{5 x + 10}$